# American Institute of Mathematical Sciences

March  2014, 7(1): 29-44. doi: 10.3934/krm.2014.7.29

## A mathematical model for value estimation with public information and herding

 1 Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received  November 2012 Revised  April 2013 Published  December 2013

This paper deals with a class of integro-differential equations modeling the dynamics of a market where agents estimate the value of a given traded good. Two basic mechanisms are assumed to concur in value estimation: interactions between agents and sources of public information and herding phenomena. A general well-posedness result is established for the initial value problem linked to the model and the asymptotic behavior in time of the related solution is characterized for some general parameter settings, which mimic different economic scenarios. Analytical results are illustrated by means of numerical simulations and lead us to conclude that, in spite of its oversimplified nature, this model is able to reproduce some emerging behaviors proper of the system under consideration. In particular, consistently with experimental evidence, the obtained results suggest that if agents are highly confident in the product, imitative and scarcely rational behaviors may lead to an over-exponential rise of the value estimated by the market, paving the way to the formation of economic bubbles.
Citation: Marcello Delitala, Tommaso Lorenzi. A mathematical model for value estimation with public information and herding. Kinetic and Related Models, 2014, 7 (1) : 29-44. doi: 10.3934/krm.2014.7.29
##### References:
 [1] G. Ajmone Marsan, N. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economic systems by functional subsystems representation, Kin. Rel. Mod., 1 (2008), 249-278. doi: 10.3934/krm.2008.1.249. [2] P. Ball, Why Society is a Complex Matter, Springer-Verlag, Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-29000-8. [3] J. Berg, M. Marsili, A. Rustichini and R. Zecchina, Statistical mechanics of asset markets with private information, J. Quant. Finance, 1 (2001), 203-211. [4] Y. Biondi, P. Giannoccolo and S. Galam, Formation of share market prices under heterogeneous beliefs and common knowledge, Physica A, 391 (2012), 5532-5545. doi: 10.1016/j.physa.2012.06.015. [5] A. Boccabella, R. Natalini and L. Pareschi, On a continuous mixed strategy model for evolutionary game-theory, Kinet. Relat. Models, 4 (2011), 187-213. doi: 10.3934/krm.2011.4.187. [6] D. Borra and T. Lorenzi, Asymptotic analysis of continuous opinion dynamics models under bounded confidence, Commun. Pure Appl. Anal., 12 (2013), 1487-1499. doi: 10.3934/cpaa.2013.12.1487. [7] J.-P. Bouchaud, Economics need a scientific revolution, Nature, 455 (2008), 1181. doi: 10.1038/4551181a. [8] J.-P. Bouchaud, The (unfortunate) complexity of the economy, Physics World, 82 (2009), 28-32. [9] L. Boudin and F. Salvarani, Modeling Opinion Formation by Means of Kinetic Equations, in Mathematical modeling of collective behavior in socio-economic and life sciences, Birkhauser, Boston, 2010. doi: 10.1007/978-0-8176-4946-3_10. [10] L. Boudin and F. Salvarani, The quasi-invariant limit for a kinetic model of sociological collective behavior, Kinet. Relat. Models, 2 (2009), 433-449. doi: 10.3934/krm.2009.2.433. [11] S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0. [12] M. Cristelli, L. Pietronero and A. Zaccaria, Critical Overview of Agent-Based Models for Economics, Proceedings of the School of Physics E. Fermi, course CLXXVI, 2010. [13] B. During, P. A. Markowich, J. F. Pietschmann and M. T. Wolfram, Boltzmann and Fokker-Planck Equations modelling Opinion Formation in the Presence of strong Leaders, Proc. Royal Soc. A, 465 (2009), 3687-3708. doi: 10.1098/rspa.2009.0239. [14] R. Gatignol, Théorie Cinétique des Gaz à Répartition Discréte des Vitèsses, (French) Lecture Notes in Physics, Vol. 36. Springer-Verlag, Berlin-New York, 1975. [15] E. Eyster and M. Rabin, naïve herding in rich-information settings, AEJ Microeconomics, 2 (2010), 221-243. doi: 10.1257/mic.2.4.221. [16] T. Kaizoji and D. Sornette, Market Bubbles and Crashes, in Encyclopedia of quantitative finance, Wiley, New York, 2010. doi: 10.1143/PTPS.162.165. [17] J.-M. Lasry and P.-L. Lions, Mean field games, Japan. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8. [18] J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, International Journal of Modern Physics C, 18 (2007), 1819-1838. doi: 10.1142/S0129183107011789. [19] T. Lux, Herd behaviour, bubbles and crashes, Economic Journal, 105 (1995), 881-896. doi: 10.2307/2235156. [20] D. Maldarella and L. Pareschi, Kinetic models for socio-economic dynamics of speculative markets, Physica A, 391 (2012), 715-730. doi: 10.1016/j.physa.2011.08.013. [21] M. Marsili, D. Challet and R. Zecchina, Exact solution of a modified El Farol's bar problem: Efficiency and the role of market impact, Physica A: Statistical Mechanics and its Applications, 280 (2000), 522-553. doi: 10.1016/S0378-4371(99)00610-X. [22] Q. Michard and J.-P. Bouchaud, Theory of collective opinion shifts: From smooth trends to abrupt swings, Eur. Phys. J. B, 47 (2005), 151-159. doi: 10.1140/epjb/e2005-00307-0. [23] B. Perthame, Trasport Equations in Biology, Frontiers in Mathematics. Birkäuser Verlag, Basel, 2007. [24] H. A. Simon, Invariants of human behavior, Annu. Rev. Psychol., 41 (1990), 1-19. [25] G. Toscani, C. Brugna and S. Demichelis, Kinetic models for the trading of goods, J. Stat. Phys., 151 (2013), 549-566. doi: 10.1007/s10955-012-0653-0.

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##### References:
 [1] G. Ajmone Marsan, N. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economic systems by functional subsystems representation, Kin. Rel. Mod., 1 (2008), 249-278. doi: 10.3934/krm.2008.1.249. [2] P. Ball, Why Society is a Complex Matter, Springer-Verlag, Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-29000-8. [3] J. Berg, M. Marsili, A. Rustichini and R. Zecchina, Statistical mechanics of asset markets with private information, J. Quant. Finance, 1 (2001), 203-211. [4] Y. Biondi, P. Giannoccolo and S. Galam, Formation of share market prices under heterogeneous beliefs and common knowledge, Physica A, 391 (2012), 5532-5545. doi: 10.1016/j.physa.2012.06.015. [5] A. Boccabella, R. Natalini and L. Pareschi, On a continuous mixed strategy model for evolutionary game-theory, Kinet. Relat. Models, 4 (2011), 187-213. doi: 10.3934/krm.2011.4.187. [6] D. Borra and T. Lorenzi, Asymptotic analysis of continuous opinion dynamics models under bounded confidence, Commun. Pure Appl. Anal., 12 (2013), 1487-1499. doi: 10.3934/cpaa.2013.12.1487. [7] J.-P. Bouchaud, Economics need a scientific revolution, Nature, 455 (2008), 1181. doi: 10.1038/4551181a. [8] J.-P. Bouchaud, The (unfortunate) complexity of the economy, Physics World, 82 (2009), 28-32. [9] L. Boudin and F. Salvarani, Modeling Opinion Formation by Means of Kinetic Equations, in Mathematical modeling of collective behavior in socio-economic and life sciences, Birkhauser, Boston, 2010. doi: 10.1007/978-0-8176-4946-3_10. [10] L. Boudin and F. Salvarani, The quasi-invariant limit for a kinetic model of sociological collective behavior, Kinet. Relat. Models, 2 (2009), 433-449. doi: 10.3934/krm.2009.2.433. [11] S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0. [12] M. Cristelli, L. Pietronero and A. Zaccaria, Critical Overview of Agent-Based Models for Economics, Proceedings of the School of Physics E. Fermi, course CLXXVI, 2010. [13] B. During, P. A. Markowich, J. F. Pietschmann and M. T. Wolfram, Boltzmann and Fokker-Planck Equations modelling Opinion Formation in the Presence of strong Leaders, Proc. Royal Soc. A, 465 (2009), 3687-3708. doi: 10.1098/rspa.2009.0239. [14] R. Gatignol, Théorie Cinétique des Gaz à Répartition Discréte des Vitèsses, (French) Lecture Notes in Physics, Vol. 36. Springer-Verlag, Berlin-New York, 1975. [15] E. Eyster and M. Rabin, naïve herding in rich-information settings, AEJ Microeconomics, 2 (2010), 221-243. doi: 10.1257/mic.2.4.221. [16] T. Kaizoji and D. Sornette, Market Bubbles and Crashes, in Encyclopedia of quantitative finance, Wiley, New York, 2010. doi: 10.1143/PTPS.162.165. [17] J.-M. Lasry and P.-L. Lions, Mean field games, Japan. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8. [18] J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, International Journal of Modern Physics C, 18 (2007), 1819-1838. doi: 10.1142/S0129183107011789. [19] T. Lux, Herd behaviour, bubbles and crashes, Economic Journal, 105 (1995), 881-896. doi: 10.2307/2235156. [20] D. Maldarella and L. Pareschi, Kinetic models for socio-economic dynamics of speculative markets, Physica A, 391 (2012), 715-730. doi: 10.1016/j.physa.2011.08.013. [21] M. Marsili, D. Challet and R. Zecchina, Exact solution of a modified El Farol's bar problem: Efficiency and the role of market impact, Physica A: Statistical Mechanics and its Applications, 280 (2000), 522-553. doi: 10.1016/S0378-4371(99)00610-X. [22] Q. Michard and J.-P. Bouchaud, Theory of collective opinion shifts: From smooth trends to abrupt swings, Eur. Phys. J. B, 47 (2005), 151-159. doi: 10.1140/epjb/e2005-00307-0. [23] B. Perthame, Trasport Equations in Biology, Frontiers in Mathematics. Birkäuser Verlag, Basel, 2007. [24] H. A. Simon, Invariants of human behavior, Annu. Rev. Psychol., 41 (1990), 1-19. [25] G. Toscani, C. Brugna and S. Demichelis, Kinetic models for the trading of goods, J. Stat. Phys., 151 (2013), 549-566. doi: 10.1007/s10955-012-0653-0.
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